And now, for your amusement and stimulation,Vampire Population EcologyIntroduction~or~A Little Math Never Hurt AnyoneWe are gathered here today to ponder the ways in which the humans and vampires ofSunnydale interact. Specifically, Betsy asked:Ooh, Brian, can you help us work out the vampire carrying capacity of a typicalpopulation? I'm assuming a typical vampire accounts for, say, 150-200 humans ayear. So how big does a town have to be to support Sunnydale's apparentlylimitless supply of vampires? Are there human warrens in the catacombssomewhere, used only for feeding purposes?The term “carrying capacity” isn’t often applied to predator population dynamics. Instead,ecologists generally estimate stable predator populations by first coming to grips with theprey’s population dynamics, including its carrying capacity. Actually, in a lot of differentcases, the prey’s carrying capacity ultimately determines how well the predator does.In principle, ecologists might employ two basic strategies to get at a problem like this. Theempiricists would go out and find a field site where they could actually observe predators andtheir prey, and just tally the results over time. The theoreticians would chuckle at theempiricists, and construct mathematical models that probably approximate the behavior ofpopulations in the field, keeping their hands more or less clean in the process.In real life, most ecologists use both strategies off and on. Unfortunately, I don’t know of anyreal life vampire populations in the field, so we’re going to have to pretend that we are stricttheoreticians. That means that we’ll be using math: some algebra, some calculus, and somematrix theory. This is O.K.! It hurts a lot less than, say, getting bitten by a vampire as you’retrying to fit the bugger with a radio collar.A ModelWhat follows is based on some of the simpler theoretical understandings of predator-preypopulation dynamics. I’m assuming that human populations are not controlled solely byvampire predation (i.e.- in the absence of vampires, the human population would stilleventually be limited by some other factor, like food supply, disease, or access to a well-written weekly news magazine. I like The Economist myself, but that’s clearly a digression).If we let H stand for the size of the human population and V stand for the size of the vampirepopulation, then we can represent the changes in each population over time with a pair ofdifferential equations:()rH K HdHdt K aHVdVbaHV mV sVdt−=−=+−where r is the intrinsic growth rate of the human population, incorporatingnatural rates of both birth and death as well as immigrationK is the human carrying capacity of the habitat in questiona is a coefficient that relates the number of human-vampireencounters to the number of actual feedingsb is the proportion of feedings in which the vampire sires the victim(i.e.- this is the vampire birth rate)m is the net rate of vampire migration into Sunnydales is the rate at which the Scoobies stake vampires (assumed to be theonly important source of vampire deaths).What we need to do, simply put, is find the equilibria that exist between these two equations.In other words, we need to find the combination(s) of human and vampire populations sizesthat satisfy both equations at the same time. As it turns out, there are three such equilibria.Without showing you the really ugly math (after all, this is a family forum), I’ll just say thattwo of the equilibria are not very interesting. They are 1) when both the humans and thevampires are completely extinct, and 2) when the vampires are extinct but the humanpopulation hovers at or near its carrying capacity. The third equilibrium is the one we careabout, wherein humans and vampires coexist. At that point, the solutions are:ˆsmHba−= andˆ1rmsVa baK−=+Notice that the actual human population doesn’t depend on carrying capacity at all, and thatthe vampire population does! yes, only an ecologist could put an exclamation point after astatement like that… We know from the existence of our other equilibria that the humanpopulation is not necessarily big enough to support a vampire population. What we need toknow is whether or not the human carrying capacity is large enough. Specifically, if K is toosmall, then1msbaK−>and the equilibrium vampire population size will be negative. Basically, the humanpopulation’s carrying capacity must be higher than its equilibrium abundance:msKba−>If this isn’t the case, then the even largest possible human population isn’t large enough, andvampires have no hope in this particular region.A TrialNow that we have a model, we can start trying out some assumptions (or, if we’re lucky,actual measurements) for the various parameters. To start with, we know from the sign in“Lover's Walk” that the human population of Sunnydale is 38,500. We also know that thetown of Berkeley, CA has a population of about 100,000. Since Berkeley is also a town witha UC campus, and is furthermore a town that has been more or less completely urbanized (thepopulation has been stable or dropping slightly for about 25 years), we will take 100,000 asthe carrying capacity for a California university town.Let’s assume the following:! Sunnydale’s human population growth rate is 10% annually, which is at the high endfor a budding California community.! A vampire feeds every three days, and encounters about one hundred potential victimsin the course of a day, meaning that 1 out of every 300 encounters involves a littlerefreshment.! An individual vampire sires a victim every other year, or once per 240 feedings.! Buffy and her Slayerettes, busy little beavers that they are, annually stake about 1/3 ofthe vampires plaguing Sunnydale.! Vampires are flocking to Sunnydale, since the Hellmouth is the underwordlyequivalent of Silicon Valley, and the demon labor market is just too good to be true.Thus, we’ll assume a yearly migration rate of about 10%, or the same as for thehumans.In terms of our model, we have:r = 0.0953a = 0.00333b = 0.00417s = 0.600K = 100000m = 0.0953note that for r, s, and m I’ve pulled a little switcheroo. In our assumptions, wespeculated as to the yearly rates of growth or migration or what-have-you. Butour model is based on a set of continuous, exponential functions, rather thandiscrete time step geometric functions. For